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Directory:Jon Awbrey/Papers/Differential Logic and Dynamic Systems 2.0 (view source)

Revision as of 02:31, 11 September 2013

4,598 bytes added ,  02:31, 11 September 2013
update
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{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:80%"
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{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"
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|+ <math>\text{Table A6.}~~\mathrm{D}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}</math>
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|+ style="height:30px" | <math>\text{Table A6.} ~~ \mathrm{D}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}\!</math>
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|- style="background:#f0f0ff"
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|- style="background:ghostwhite"
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| width="10%" | &nbsp;
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| style="width:10%; border-bottom:1px solid black" | &nbsp;
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| width="18%" | <math>f\!</math>
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| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math>
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| width="18%" | <math>\mathrm{D}f|_{xy}\!</math>
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| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |
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| width="18%" | <math>\mathrm{D}f|_{x(y)}\!</math>
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<math>\mathrm{D}f|_{xy}\!</math>
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| width="18%" | <math>\mathrm{D}f|_{(x)y}~\!</math>
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| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
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| width="18%" | <math>\mathrm{D}f|_{(x)(y)}\!</math>
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<math>\mathrm{D}f|_{x \texttt{(} y \texttt{)}}\!</math>
 +
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
 +
<math>\mathrm{D}f|_{\texttt{(} x \texttt{)} y}\!</math>
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| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |
 +
<math>\mathrm{D}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}\!</math>
 
|-
 
|-
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| <math>f_{0}\!</math>
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| style="border-top:1px solid black" | <math>f_{0}\!</math>
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| <math>(~)\!</math>
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| style="border-top:1px solid black; border-left:1px solid black"  | <math>0\!</math>
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| <math>(~)\!</math>
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| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math>
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| <math>(~)\!</math>
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| style="border-top:1px solid black; border-left:1px solid black"  | <math>0\!</math>
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| <math>(~)\!</math>
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| style="border-top:1px solid black; border-left:1px solid black"  | <math>0\!</math>
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| <math>(~)\!</math>
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| style="border-top:1px solid black; border-left:1px solid black"  | <math>0\!</math>
 
|-
 
|-
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|
+
| style="border-top:1px solid black" | <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math>
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| style="border-top:1px solid black; border-left:1px solid black" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
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f_1
+
\texttt{(} x \texttt{)(} y \texttt{)}
 
\\
 
\\
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f_2
+
\texttt{(} x \texttt{)~} y \texttt{~}
 
\\
 
\\
−
f_4
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\texttt{~} x \texttt{~(} y \texttt{)}
 
\\
 
\\
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f_8
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\texttt{~} x \texttt{~~} y \texttt{~}
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\end{matrix}\!</math>
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\end{matrix}</math>
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| style="border-top:1px solid black; border-left:4px double black" |
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<math>\begin{matrix}
  −
(x)(y)
  −
\\
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(x)~y~
  −
\\
  −
~x~(y)
  −
\\
  −
~x~~y~
  −
\end{matrix}\!</math>
  −
|
   
<math>\begin{matrix}
 
<math>\begin{matrix}
−
~~\mathrm{d}x~~\mathrm{d}y~~
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\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}
 
\\
 
\\
−
~~\mathrm{d}x~(\mathrm{d}y)~
+
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}
 
\\
 
\\
−
~(\mathrm{d}x)~\mathrm{d}y~~
+
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}
 
\\
 
\\
−
((\mathrm{d}x)(\mathrm{d}y))
+
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
−
\end{matrix}\!</math>
+
\end{matrix}</math>
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|
+
| style="border-top:1px solid black; border-left:1px solid black" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
~~\mathrm{d}x~(\mathrm{d}y)~
+
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}
 
\\
 
\\
−
~~\mathrm{d}x~~\mathrm{d}y~~
+
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}
 
\\
 
\\
−
((\mathrm{d}x)(\mathrm{d}y))
+
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
 
\\
 
\\
−
~(\mathrm{d}x)~\mathrm{d}y~~
+
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}
 
\end{matrix}\!</math>
 
\end{matrix}\!</math>
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|
+
| style="border-top:1px solid black; border-left:1px solid black" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
~(\mathrm{d}x)~\mathrm{d}y~~
+
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}
 
\\
 
\\
−
((\mathrm{d}x)(\mathrm{d}y))
+
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
 
\\
 
\\
−
~~\mathrm{d}x~~\mathrm{d}y~~
+
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}
 
\\
 
\\
−
~~\mathrm{d}x~(\mathrm{d}y)~
+
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}
−
\end{matrix}\!</math>
+
\end{matrix}</math>
−
|
+
| style="border-top:1px solid black; border-left:1px solid black" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
((\mathrm{d}x)(\mathrm{d}y))
+
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
 
\\
 
\\
−
~(\mathrm{d}x)~\mathrm{d}y~~
+
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}
 
\\
 
\\
−
~~\mathrm{d}x~(\mathrm{d}y)~
+
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}
 
\\
 
\\
−
~~\mathrm{d}x~~\mathrm{d}y~~
+
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}
−
\end{matrix}\!</math>
+
\end{matrix}</math>
 
|-
 
|-
−
|
+
| style="border-top:1px solid black" | <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math>
−
<math>\begin{matrix}
+
| style="border-top:1px solid black; border-left:1px solid black" |
−
f_3
+
<math>\begin{matrix}\texttt{(} x \texttt{)}\\\texttt{~} x \texttt{~}\end{matrix}</math>
−
\\
+
| style="border-top:1px solid black; border-left:4px double black" |
−
f_{12}
+
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math>
−
\end{matrix}\!</math>
+
| style="border-top:1px solid black; border-left:1px solid black" |
−
|
+
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math>
−
<math>\begin{matrix}
+
| style="border-top:1px solid black; border-left:1px solid black" |
−
(x)
+
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math>
−
\\
+
| style="border-top:1px solid black; border-left:1px solid black" |
−
~x~
+
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math>
−
\end{matrix}\!</math>
  −
|
  −
<math>\begin{matrix}
  −
\mathrm{d}x
  −
\\
  −
\mathrm{d}x
  −
\end{matrix}\!</math>
  −
|
  −
<math>\begin{matrix}
  −
\mathrm{d}x
  −
\\
  −
\mathrm{d}x
  −
\end{matrix}\!</math>
  −
|
  −
<math>\begin{matrix}
  −
\mathrm{d}x
  −
\\
  −
\mathrm{d}x
  −
\end{matrix}\!</math>
  −
|
  −
<math>\begin{matrix}
  −
\mathrm{d}x
  −
\\
  −
\mathrm{d}x
  −
\end{matrix}\!</math>
   
|-
 
|-
−
|
+
| style="border-top:1px solid black" | <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math>
−
<math>\begin{matrix}
+
| style="border-top:1px solid black; border-left:1px solid black" |
−
f_6
  −
\\
  −
f_9
  −
\end{matrix}\!</math>
  −
|
   
<math>\begin{matrix}
 
<math>\begin{matrix}
−
~(x,~y)~
+
\texttt{~(} x \texttt{,~} y \texttt{)~}
 
\\
 
\\
−
((x,~y))
+
\texttt{((} x \texttt{,~} y \texttt{))}
−
\end{matrix}\!</math>
+
\end{matrix}</math>
−
|
+
| style="border-top:1px solid black; border-left:4px double black" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
(\mathrm{d}x,~\mathrm{d}y)
+
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}
 
\\
 
\\
−
(\mathrm{d}x,~\mathrm{d}y)
+
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}
−
\end{matrix}\!</math>
+
\end{matrix}</math>
−
|
+
| style="border-top:1px solid black; border-left:1px solid black" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
(\mathrm{d}x,~\mathrm{d}y)
+
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}
 
\\
 
\\
−
(\mathrm{d}x,~\mathrm{d}y)
+
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}
−
\end{matrix}\!</math>
+
\end{matrix}</math>
−
|
+
| style="border-top:1px solid black; border-left:1px solid black" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
(\mathrm{d}x,~\mathrm{d}y)
+
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}
 
\\
 
\\
−
(\mathrm{d}x,~\mathrm{d}y)
+
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}
−
\end{matrix}\!</math>
+
\end{matrix}</math>
−
|
+
| style="border-top:1px solid black; border-left:1px solid black" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
(\mathrm{d}x,~\mathrm{d}y)
+
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}
 
\\
 
\\
−
(\mathrm{d}x,~\mathrm{d}y)
+
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}
−
\end{matrix}\!</math>
+
\end{matrix}</math>
 
|-
 
|-
−
|
+
| style="border-top:1px solid black" | <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math>
−
<math>\begin{matrix}
+
| style="border-top:1px solid black; border-left:1px solid black" |
−
f_5
  −
\\
  −
f_{10}
  −
\end{matrix}\!</math>
  −
|
  −
<math>\begin{matrix}
  −
(y)
  −
\\
  −
~y~
  −
\end{matrix}\!</math>
  −
|
  −
<math>\begin{matrix}
  −
\mathrm{d}y
  −
\\
  −
\mathrm{d}y
  −
\end{matrix}\!</math>
  −
|
  −
<math>\begin{matrix}
  −
\mathrm{d}y
  −
\\
  −
\mathrm{d}y
  −
\end{matrix}\!</math>
  −
|
  −
<math>\begin{matrix}
  −
\mathrm{d}y
  −
\\
  −
\mathrm{d}y
  −
\end{matrix}\!</math>
  −
|
   
<math>\begin{matrix}
 
<math>\begin{matrix}
−
\mathrm{d}y
+
\texttt{(} y \texttt{)}
 
\\
 
\\
−
\mathrm{d}y
+
\texttt{~} y \texttt{~}
−
\end{matrix}\!</math>
+
\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:4px double black" |
 +
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 +
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math>
 
|-
 
|-
−
|
+
| style="border-top:1px solid black" | <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math>
 +
| style="border-top:1px solid black; border-left:1px solid black" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
f_7
+
\texttt{(~} x \texttt{~~} y \texttt{~)}
 
\\
 
\\
−
f_{11}
+
\texttt{(~} x \texttt{~(} y \texttt{))}
 
\\
 
\\
−
f_{13}
+
\texttt{((} x \texttt{)~} y \texttt{~)}
 
\\
 
\\
−
f_{14}
+
\texttt{((} x \texttt{)(} y \texttt{))}
−
\end{matrix}\!</math>
+
\end{matrix}</math>
−
|
+
| style="border-top:1px solid black; border-left:4px double black" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
(~x~~y~)
+
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
 
\\
 
\\
−
(~x~(y))
+
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}
 
\\
 
\\
−
((x)~y~)
+
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}
 
\\
 
\\
−
((x)(y))
+
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}
−
\end{matrix}\!</math>
+
\end{matrix}</math>
−
|
+
| style="border-top:1px solid black; border-left:1px solid black" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
((\mathrm{d}x)(\mathrm{d}y))
+
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}
 
\\
 
\\
−
~(\mathrm{d}x)~\mathrm{d}y~~
+
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
 
\\
 
\\
−
~~\mathrm{d}x~(\mathrm{d}y)~
+
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}
 
\\
 
\\
−
~~\mathrm{d}x~~\mathrm{d}y~~
+
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}
−
\end{matrix}\!</math>
+
\end{matrix}</math>
−
|
+
| style="border-top:1px solid black; border-left:1px solid black" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
~(\mathrm{d}x)~\mathrm{d}y~~
+
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}
 
\\
 
\\
−
((\mathrm{d}x)(\mathrm{d}y))
+
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}
 
\\
 
\\
−
~~\mathrm{d}x~~\mathrm{d}y~~
+
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
 
\\
 
\\
−
~~\mathrm{d}x~(\mathrm{d}y)~
+
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}
 
\end{matrix}\!</math>
 
\end{matrix}\!</math>
−
|
+
| style="border-top:1px solid black; border-left:1px solid black" |
 
<math>\begin{matrix}
 
<math>\begin{matrix}
−
~~\mathrm{d}x~(\mathrm{d}y)~
+
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}
−
\\
  −
~~\mathrm{d}x~~\mathrm{d}y~~
   
\\
 
\\
−
((\mathrm{d}x)(\mathrm{d}y))
+
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}
 
\\
 
\\
−
~(\mathrm{d}x)~\mathrm{d}y~~
+
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}
−
\end{matrix}\!</math>
  −
|
  −
<math>\begin{matrix}
  −
~~\mathrm{d}x~~\mathrm{d}y~~
   
\\
 
\\
−
~~\mathrm{d}x~(\mathrm{d}y)~
+
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}
−
\\
+
\end{matrix}</math>
−
~(\mathrm{d}x)~\mathrm{d}y~~
  −
\\
  −
((\mathrm{d}x)(\mathrm{d}y))
  −
\end{matrix}\!</math>
   
|-
 
|-
−
| <math>f_{15}\!</math>
+
| style="border-top:1px solid black" | <math>f_{15}\!</math>
−
| <math>((~))\!</math>
+
| style="border-top:1px solid black; border-left:1px solid black"  | <math>1\!</math>
−
| <math>(~)\!</math>
+
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math>
−
| <math>(~)\!</math>
+
| style="border-top:1px solid black; border-left:1px solid black"  | <math>0\!</math>
−
| <math>(~)\!</math>
+
| style="border-top:1px solid black; border-left:1px solid black"  | <math>0\!</math>
−
| <math>(~)\!</math>
+
| style="border-top:1px solid black; border-left:1px solid black"  | <math>0\!</math>
 
|}
 
|}
  
Jon Awbrey
12,080

edits

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